Z cyfr 1-9 utworzono trzy liczby trzycyfrowe, wśród których żadna cyfra się nie powtarza (czyli np. 132, 467 i 598 - wszystkie cyfry różne). Suma dwóch liczb jest równa trzeciej.
Podaj wszystkie możliwe liczby (trójki liczb) spełniające powyższe warunki.
ABC+DEF=GHI
-
athame
- Użytkownik
- Posty: 558
- Rejestracja: 2 lut 2012, o 21:42
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- Lokalizacja: Radom
- Podziękował: 1 raz
- Pomógł: 64 razy
ABC+DEF=GHI
Poniżej wszystkie rozwiązania:
\(\displaystyle{ 124+659=783}\)
\(\displaystyle{ 125+739=864}\)
\(\displaystyle{ 127+359=486}\)
\(\displaystyle{ 127+368=495}\)
\(\displaystyle{ 128+367=495}\)
\(\displaystyle{ 128+439=567}\)
\(\displaystyle{ 129+357=486}\)
\(\displaystyle{ 129+438=567}\)
\(\displaystyle{ 129+654=783}\)
\(\displaystyle{ 129+735=864}\)
\(\displaystyle{ 134+658=792}\)
\(\displaystyle{ 135+729=864}\)
\(\displaystyle{ 138+429=567}\)
\(\displaystyle{ 138+654=792}\)
\(\displaystyle{ 139+428=567}\)
\(\displaystyle{ 139+725=864}\)
\(\displaystyle{ 142+596=738}\)
\(\displaystyle{ 142+695=837}\)
\(\displaystyle{ 143+586=729}\)
\(\displaystyle{ 145+692=837}\)
\(\displaystyle{ 146+583=729}\)
\(\displaystyle{ 146+592=738}\)
\(\displaystyle{ 152+487=639}\)
\(\displaystyle{ 152+784=936}\)
\(\displaystyle{ 154+629=783}\)
\(\displaystyle{ 154+638=792}\)
\(\displaystyle{ 154+782=936}\)
\(\displaystyle{ 157+329=486}\)
\(\displaystyle{ 157+482=639}\)
\(\displaystyle{ 158+634=792}\)
\(\displaystyle{ 159+327=486}\)
\(\displaystyle{ 159+624=783}\)
\(\displaystyle{ 162+387=549}\)
\(\displaystyle{ 162+783=945}\)
\(\displaystyle{ 163+782=945}\)
\(\displaystyle{ 167+328=495}\)
\(\displaystyle{ 167+382=549}\)
\(\displaystyle{ 168+327=495}\)
\(\displaystyle{ 173+286=459}\)
\(\displaystyle{ 173+295=468}\)
\(\displaystyle{ 175+293=468}\)
\(\displaystyle{ 176+283=459}\)
\(\displaystyle{ 182+367=549}\)
\(\displaystyle{ 182+394=576}\)
\(\displaystyle{ 182+457=639}\)
\(\displaystyle{ 182+493=675}\)
\(\displaystyle{ 182+754=936}\)
\(\displaystyle{ 182+763=945}\)
\(\displaystyle{ 183+276=459}\)
\(\displaystyle{ 183+492=675}\)
\(\displaystyle{ 183+546=729}\)
\(\displaystyle{ 183+762=945}\)
\(\displaystyle{ 184+392=576}\)
\(\displaystyle{ 184+752=936}\)
\(\displaystyle{ 186+273=459}\)
\(\displaystyle{ 186+543=729}\)
\(\displaystyle{ 187+362=549}\)
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\(\displaystyle{ 192+384=576}\)
\(\displaystyle{ 192+483=675}\)
\(\displaystyle{ 192+546=738}\)
\(\displaystyle{ 192+645=837}\)
\(\displaystyle{ 193+275=468}\)
\(\displaystyle{ 193+482=675}\)
\(\displaystyle{ 194+382=576}\)
\(\displaystyle{ 195+273=468}\)
\(\displaystyle{ 195+642=837}\)
\(\displaystyle{ 196+542=738}\)
\(\displaystyle{ 214+569=783}\)
\(\displaystyle{ 214+659=873}\)
\(\displaystyle{ 215+478=693}\)
\(\displaystyle{ 215+748=963}\)
\(\displaystyle{ 216+378=594}\)
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\(\displaystyle{ 218+349=567}\)
\(\displaystyle{ 218+376=594}\)
\(\displaystyle{ 218+439=657}\)
\(\displaystyle{ 218+475=693}\)
\(\displaystyle{ 218+736=954}\)
\(\displaystyle{ 218+745=963}\)
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\(\displaystyle{ 219+438=657}\)
\(\displaystyle{ 219+564=783}\)
\(\displaystyle{ 219+654=873}\)
\(\displaystyle{ 234+657=891}\)
\(\displaystyle{ 235+746=981}\)
\(\displaystyle{ 236+718=954}\)
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\(\displaystyle{ 237+654=891}\)
\(\displaystyle{ 238+419=657}\)
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\(\displaystyle{ 241+596=837}\)
\(\displaystyle{ 243+576=819}\)
\(\displaystyle{ 243+675=918}\)
\(\displaystyle{ 245+673=918}\)
\(\displaystyle{ 245+718=963}\)
\(\displaystyle{ 245+736=981}\)
\(\displaystyle{ 246+573=819}\)
\(\displaystyle{ 246+591=837}\)
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\(\displaystyle{ 248+319=567}\)
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\(\displaystyle{ 251+397=648}\)
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\(\displaystyle{ 273+186=459}\)
\(\displaystyle{ 273+195=468}\)
\(\displaystyle{ 273+546=819}\)
\(\displaystyle{ 273+591=864}\)
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\(\displaystyle{ 273+681=954}\)
\(\displaystyle{ 275+193=468}\)
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\(\displaystyle{ 276+183=459}\)
\(\displaystyle{ 276+318=594}\)
\(\displaystyle{ 276+543=819}\)
\(\displaystyle{ 278+316=594}\)
\(\displaystyle{ 278+415=693}\)
\(\displaystyle{ 281+394=675}\)
\(\displaystyle{ 281+673=954}\)
\(\displaystyle{ 283+176=459}\)
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\(\displaystyle{ 284+391=675}\)
\(\displaystyle{ 286+173=459}\)
\(\displaystyle{ 291+357=648}\)
\(\displaystyle{ 291+384=675}\)
\(\displaystyle{ 291+546=837}\)
\(\displaystyle{ 291+573=864}\)
\(\displaystyle{ 293+175=468}\)
\(\displaystyle{ 293+571=864}\)
\(\displaystyle{ 294+381=675}\)
\(\displaystyle{ 295+173=468}\)
\(\displaystyle{ 296+541=837}\)
\(\displaystyle{ 297+351=648}\)
\(\displaystyle{ 314+658=972}\)
\(\displaystyle{ 316+278=594}\)
\(\displaystyle{ 317+529=846}\)
\(\displaystyle{ 317+628=945}\)
\(\displaystyle{ 318+249=567}\)
\(\displaystyle{ 318+276=594}\)
\(\displaystyle{ 318+627=945}\)
\(\displaystyle{ 318+654=972}\)
\(\displaystyle{ 319+248=567}\)
\(\displaystyle{ 319+527=846}\)
\(\displaystyle{ 324+567=891}\)
\(\displaystyle{ 324+657=981}\)
\(\displaystyle{ 327+159=486}\)
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\(\displaystyle{ 341+586=927}\)
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\(\displaystyle{ 376+218=594}\)
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\(\displaystyle{ 384+291=675}\)
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\(\displaystyle{ 418+239=657}\)
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\(\displaystyle{ 428+139=567}\)
\(\displaystyle{ 429+138=567}\)
\(\displaystyle{ 438+129=567}\)
\(\displaystyle{ 438+219=657}\)
\(\displaystyle{ 439+128=567}\)
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\(\displaystyle{ 452+187=639}\)
\(\displaystyle{ 452+367=819}\)
\(\displaystyle{ 457+182=639}\)
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\(\displaystyle{ 462+357=819}\)
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\(\displaystyle{ 586+143=729}\)
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\(\displaystyle{ 614+259=873}\)
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\(\displaystyle{ 182+493=675}\)
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\(\displaystyle{ 182+763=945}\)
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\(\displaystyle{ 183+492=675}\)
\(\displaystyle{ 183+546=729}\)
\(\displaystyle{ 183+762=945}\)
\(\displaystyle{ 184+392=576}\)
\(\displaystyle{ 184+752=936}\)
\(\displaystyle{ 186+273=459}\)
\(\displaystyle{ 186+543=729}\)
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\(\displaystyle{ 192+384=576}\)
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\(\displaystyle{ 214+659=873}\)
\(\displaystyle{ 215+478=693}\)
\(\displaystyle{ 215+748=963}\)
\(\displaystyle{ 216+378=594}\)
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\(\displaystyle{ 218+439=657}\)
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\(\displaystyle{ 415+278=693}\)
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\(\displaystyle{ 452+187=639}\)
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\(\displaystyle{ 492+183=675}\)
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\(\displaystyle{ 524+367=891}\)
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\(\displaystyle{ 541+296=837}\)
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\(\displaystyle{ 543+186=729}\)
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\(\displaystyle{ 564+219=783}\)
\(\displaystyle{ 564+327=891}\)
\(\displaystyle{ 567+324=891}\)
\(\displaystyle{ 569+214=783}\)
\(\displaystyle{ 571+293=864}\)
\(\displaystyle{ 572+346=918}\)
\(\displaystyle{ 573+246=819}\)
\(\displaystyle{ 573+291=864}\)
\(\displaystyle{ 576+243=819}\)
\(\displaystyle{ 576+342=918}\)
\(\displaystyle{ 581+346=927}\)
\(\displaystyle{ 583+146=729}\)
\(\displaystyle{ 586+143=729}\)
\(\displaystyle{ 586+341=927}\)
\(\displaystyle{ 591+246=837}\)
\(\displaystyle{ 591+273=864}\)
\(\displaystyle{ 592+146=738}\)
\(\displaystyle{ 593+271=864}\)
\(\displaystyle{ 596+142=738}\)
\(\displaystyle{ 596+241=837}\)
\(\displaystyle{ 614+259=873}\)
\(\displaystyle{ 614+358=972}\)
\(\displaystyle{ 617+328=945}\)
\(\displaystyle{ 618+327=945}\)
\(\displaystyle{ 618+354=972}\)
\(\displaystyle{ 619+254=873}\)
\(\displaystyle{ 624+159=783}\)
\(\displaystyle{ 624+357=981}\)
\(\displaystyle{ 627+318=945}\)
\(\displaystyle{ 627+354=981}\)
\(\displaystyle{ 628+317=945}\)
\(\displaystyle{ 629+154=783}\)
\(\displaystyle{ 634+158=792}\)
\(\displaystyle{ 634+257=891}\)
\(\displaystyle{ 637+254=891}\)
\(\displaystyle{ 638+154=792}\)
\(\displaystyle{ 642+195=837}\)
\(\displaystyle{ 643+275=918}\)
\(\displaystyle{ 645+192=837}\)
\(\displaystyle{ 645+273=918}\)
\(\displaystyle{ 654+129=783}\)
\(\displaystyle{ 654+138=792}\)
\(\displaystyle{ 654+219=873}\)
\(\displaystyle{ 654+237=891}\)
\(\displaystyle{ 654+318=972}\)
\(\displaystyle{ 654+327=981}\)
\(\displaystyle{ 657+234=891}\)
\(\displaystyle{ 657+324=981}\)
\(\displaystyle{ 658+134=792}\)
\(\displaystyle{ 658+314=972}\)
\(\displaystyle{ 659+124=783}\)
\(\displaystyle{ 659+214=873}\)
\(\displaystyle{ 671+283=954}\)
\(\displaystyle{ 673+245=918}\)
\(\displaystyle{ 673+281=954}\)
\(\displaystyle{ 675+243=918}\)
\(\displaystyle{ 681+273=954}\)
\(\displaystyle{ 683+271=954}\)
\(\displaystyle{ 692+145=837}\)
\(\displaystyle{ 695+142=837}\)
\(\displaystyle{ 715+248=963}\)
\(\displaystyle{ 716+238=954}\)
\(\displaystyle{ 718+236=954}\)
\(\displaystyle{ 718+245=963}\)
\(\displaystyle{ 725+139=864}\)
\(\displaystyle{ 729+135=864}\)
\(\displaystyle{ 735+129=864}\)
\(\displaystyle{ 735+246=981}\)
\(\displaystyle{ 736+218=954}\)
\(\displaystyle{ 736+245=981}\)
\(\displaystyle{ 738+216=954}\)
\(\displaystyle{ 739+125=864}\)
\(\displaystyle{ 745+218=963}\)
\(\displaystyle{ 745+236=981}\)
\(\displaystyle{ 746+235=981}\)
\(\displaystyle{ 748+215=963}\)
\(\displaystyle{ 752+184=936}\)
\(\displaystyle{ 754+182=936}\)
\(\displaystyle{ 762+183=945}\)
\(\displaystyle{ 763+182=945}\)
\(\displaystyle{ 782+154=936}\)
\(\displaystyle{ 782+163=945}\)
\(\displaystyle{ 783+162=945}\)
\(\displaystyle{ 784+152=936}\)
Ostatnio zmieniony 30 wrz 2016, o 16:14 przez athame, łącznie zmieniany 1 raz.
-
rafaluk
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- Podziękował: 114 razy
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ABC+DEF=GHI
Ojej, sporo
Spodziewałem się kilku wyników... poza tym nie ukrywam, że w takich zadaniach o wiele bardziej satysfakcjonująca jest matematyczna metoda rozwiązania, a nie gotowy wynik. Domyślam się, że napisałeś prosty algorytm (pętla od 123456789 do 987654321, dla każdej liczby najpierw sprawdzenie czy są różne cyfry i jeśli tak to rozbicie na 3 liczby i czy spełnione jest dodawanie).
Spodziewałem się kilku wyników... poza tym nie ukrywam, że w takich zadaniach o wiele bardziej satysfakcjonująca jest matematyczna metoda rozwiązania, a nie gotowy wynik. Domyślam się, że napisałeś prosty algorytm (pętla od 123456789 do 987654321, dla każdej liczby najpierw sprawdzenie czy są różne cyfry i jeśli tak to rozbicie na 3 liczby i czy spełnione jest dodawanie).
-
athame
- Użytkownik
- Posty: 558
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ABC+DEF=GHI
Sporo - \(\displaystyle{ 336}\) rozwiązania - ale wszystko poprawnie. Chętnie zobaczę metodę matematyczną. Mój algorytm działa inaczej niż przytoczyłeś, ale też jest brute force (lekko ograniczony).
-
athame
- Użytkownik
- Posty: 558
- Rejestracja: 2 lut 2012, o 21:42
- Płeć: Mężczyzna
- Lokalizacja: Radom
- Podziękował: 1 raz
- Pomógł: 64 razy
ABC+DEF=GHI
Zgadza się, dodawanie jest przemienne, więc wyników nieuwzględniających kolejności występowania liczb będzie dwa razy mniej. Ja wypisałem wszystkie możliwe rozwiązania, zakładając że kolejność jest istotna. Odsianie niepotrzebnych równań jest bardzo proste, nadal jednak nie wiem jak można rozwiązać to zadanie metodami "sprytnymi".